When a tree falls on an improved structure in one’s property an insurance claim may be made. In my experience the policy holder insures the house, not the trees. So when a tree in your yard falls down and hits your neighbor’s house/car/shed, it is the responsibility of the homeowners to pick up the tab. Same goes for you too. During storm related instances we see this a lot.
There has been a few exceptions to the rule but most of the time insurance companies will only want to pay to have the tree removed off of the structure and to have the structure repaired.